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If also A has a unit element e and is not the zero algebra (in other words,
if e 7^ 0) we shall always assume that the norm satisfies the additional condi-

(15.1.2)                                                        \\e\\ = 1.

The inequality (15.1.1) shows that the bilinear mapping (x, y)\-+xy of
A x A into A is continuous (5.5.1).

A complete normed algebra (i.e., a normed algebra in which the under-
lying normed vector space is a Banach space) is called a Banach algebra.

It is clear that every subalgebra B of a normed algebra A (if A has a unit
element e, we require that B contains e), endowed with the restriction to B of
the norm ||jc||, is a normed algebra. If m is a closed two-sided ideal of A, then
the quotient algebra A/m, endowed with the norm ( induced by
||jc||, is again a normed algebra. For if x, y are two elements of A/m, then for
each e > 0 there exist x e x and y e y such that

from which it follows that ||jcy|| ^ (||jc|| + fi)(||j>|| + e); since xy e xy and since
e was arbitrary, this shows that \\xy\\ rg \\x\\ - \\y\\. If moreover A has a unit
element e, and if m + A, then e is the unit element of A/m and is ^0, hence
\\e\\ g \\e\\ = 1 by definition, and on the other hand \\e\\ = \\e2\\ ^ \\e\\2, which
implies that \\e\\ ^ 1. Hence \\e\\ = 1, and so A/m is a normed algebra.

(15.1.3) Let A be a normed algebra. Then the closure in A of a subalgebra
of A (resp. of a commutative subalgebra, a left ideal, a right ideal) is a sub-
algebra (resp. a commutative subalgebra, a left ideal, a right ideal).

By virtue of (5.4.1) and the principle of extension of identities, this follows
directly from the continuity of multiplication in A, by the same proof as
in (5.4.1).

If A, B are two normed algebras, an algebra isomorphism u : A -+ B is
said to be a topological isomorphism if it is bicontinuous, that is if (5.5.1 ) there
exist two real numbers a > 0 and b > 0 such that a \\x\\ <| ||H(JC)|| ^ b \\x\\ for
all x e A (5.5.1 ). The isomorphism u is said to be isometric if in addition we
have ||M(JC)|| = \\x\\ for all xe A.pectral theory;